Oh Yes!. That old problem where 12 volts just spiking the starter solenoid. Reminds me of a 1960 Chev. truck starter solenoid. Copper contacts need replacing.I don't believe it is a safety switch as they generally do not allow the engine to turn over at all. It sounds like the engine is not getting current from the starter switch or maybe a solenoid. That starter solenoid sounds like it is bad.
So, remove the starter solenoid... probably starter and solenoid is in one unit. Also, you need to look at the cable from battery to solenoid for corrosion.